Abstract

Consider longitudinal networks whose edges turn on and off according to a discrete-time Markov chain with exponential-family transition probabilities. We characterize when their joint distributions are also exponential families with the same parameter, and show that the permutation-uniform subclass of these chains permit interpretation as an independent, identically distributed sequence on the same state space. We apply these ideas to temporal exponential random graph models, for which permutation uniformity is well suited, and discuss mean-parameter convergence, dyadic independence, and exchangeability. The framework facilitates applying standard tools to longitudinal-network Markov chains from either asymptotics or single-observation exponential random graph models.
The latter are often in log-linear form, allowing us to frame the problem of testing model fit through an exact conditional test whose p-value can be approximated efficiently in networks of both small and moderately large sizes. An important extension of this theory is to latent-variable blockmodels, an application which will be briefly discussed.

This talk is based on joint work with William K. Schwartz, Hemanshu Kaul, Despina Stasi, Elizabeth Gross, Debdeep Pati, and Vishesh Karwa.